It is understood the couple were at Windsor Castle on Thursday.
The trip to the UK comes after Harry was absent from the Duke of Edinburgh’s memorial service in London last month.
He is bringing a claim against the Home Office after being told he would no longer be given the same degree of personal protective security when visiting from the US, despite offering to pay for it himself.
Harry wants to bring his children to visit, but he and his family are “unable to return to his home” because it is too dangerous, his legal representative has said.
A representative for Harry said earlier in March that he would miss the service for his grandfather, but that he hoped to visit the Queen soon.
On Thursday night, a representative for the couple confirmed they visited the Queen ahead of their appearance at the Invictus Games, which open this weekend.
Joe Little, managing editor of Majesty Magazine, described it as “a very welcome, if long overdue move” on the part of Harry and Meghan.
This week Dai Davies, a former head of royal protection at Scotland Yard, said that the idea Harry is unable to come to the UK due to safety concerns is “sheer unadulterated nonsense”.
He said the duke was “not an expert in security”.
“The truth is, I have no doubt that he will be protected, whether he takes his own team or whether the Dutch authorities give him some kind of VIP status," Mr Davies said of the games in The Hague.
“The truth is … both countries, ours and the Netherlands, have sophisticated systems of determining risk.
“This country has determined he is no longer at risk. That may or may not be true but people far better qualified than I am now can make that decision and they never make it lightly.”
He said the Ministry of Defence was involved in looking after royals in the Netherlands, and that the Dutch royal protection squad was not as big or as experienced as the UK’s.
Meghan’s attendance at the Invictus Games, which will be for the first few days, was confirmed by their representative on Monday.
Her trip to the UK and the Netherlands is her first to Europe since 2020.
The couple are expected to make an appearance on the “yellow carpet” at a friends and family reception at the Invictus Games on Friday.
A film crew is expected to be joining the couple at the games as part of filming for a Netflix series called Heart of Invictus.
The Invictus Games was where Harry and Meghan chose to make their first public appearance together almost five years ago.
Meghan attended the Invictus Games opening ceremony in Toronto, Canada, on September 24, 2017, but she sat about 18 seats apart from Harry.
The following day, the pair emerged hand in hand to make their first official public appearance together at the wheelchair tennis.
Harry founded the games to aid the rehabilitation of injured or sick military personnel and veterans from around the globe, giving them the challenge of competing in sporting events similar to the Paralympics.